Question

A vessel at rest explodes, breaking into three pieces. Two pieces, one with twice the mass of the other, fly off perpendicular to one another with the same speed of \(31.4 \mathrm{~m} / \mathrm{s}\). The third piece has three times the mass of the lightest piece. Find the magnitude and direction of its velocity immediately after the explosion. (Specify the direction by giving the angle from the line of travel of the least massive piece.)

Short answer

The magnitude of the velocity of the third piece is \(31.4 \, m/s\) and it moves in a direction \(180\) degrees from the line of travel of the lightest piece.

Step-by-step solution

Determine the mass of each piece

Let's denote the mass of the lightest piece as \(m\). Therefore, other two pieces have masses of \(2m\) and \(3m\). The speed for the first two pieces is given as \(31.4 \, m/s\). Although the problem states the third piece also flies off, it does not specify its speed or direction. That's what we need to find.

Calculate Sum of Momenta

The total momentum of the system is the vector sum of individual momenta of the three pieces. The momentum (denoted by \(\vec{p}\)) can be calculated with the formula \(p = m \cdot v\), where \(m\) is the mass and \(v\) is the velocity of the object. The first piece has momentum \(31.4m\), the second piece has momentum \(62.8m\). Since there is no external force acting on the object, the total momentum after the explosion should be zero.

Calculate the velocity of the third piece

In order to maintain the total momentum as zero, the momentum of the third piece must be equal to the sum of the momenta of the first two pieces (but in the opposite direction), i.e., \((31.4m + 62.8m) = -94.2m\). Dividing this by the mass of the third piece \(3m\), we find the speed of the third piece is \(-94.2m/(3m) = -31.4 \, m/s\).

Determine the direction of the third piece

The third piece moves in the opposite direction to the sum of the momenta of the two other pieces. This is perpendicular to the direction of the piece\(2m\) and opposite to the direction of \(m\), which the angle from the line of travel of the least massive piece is \(180\) degrees.

Key concepts

Linear Momentum

Linear momentum is a measure of an object's motion, combining its mass and velocity into a single quantity represented as \textbf{\(\boldsymbol{\boldsymbol{\boldsymbol{p}}} = m \times v\)}. In the context of an explosion problem, such as a vessel at rest breaking into pieces, the total linear momentum before and after the event remains constant if no external forces act on the system.

This principle, known as the conservation of linear momentum, allows us to solve for unknowns when an object breaks apart. Since the vessel in the exercise was initially at rest, its total momentum was zero. Following the explosion, the momentum of all pieces, when vectorially added, must also result in zero to satisfy conservation laws. Hence, the directions and magnitudes of the velocities of the pieces are constrained by the mass and velocity of each fragment.

Explosion Problem

An explosion problem involves an object that breaks apart into multiple pieces spontaneously. The key to solving such a problem lies in the concept of momentum conservation. Before the explosion, the object is typically at rest, and hence, the total momentum of the system is zero. After the explosion, despite pieces flying off in different directions, the vector sum of their momenta must still be zero.

With the information given for two of the pieces in our exercise (masses and velocities), we can deduce the momentum of the third piece so that the system adheres to momentum conservation. This sort of problem is an excellent exercise in both applying the conservation laws of physics and in handling vector quantities that have direction and magnitude.

Vector Summation

Vector summation is the process of adding two or more vectors together to find a resultant vector. Vectors have both magnitude and direction, which makes their sum different from just adding numerical values. In physics problems, especially those involving momenta, it is crucial to consider the direction of each vector.

To sum vectors, one must combine the horizontal components and then the vertical components separately. If the vectors are perpendicular, as in our example with the pieces flying off at right angles, the summing process simplifies, because their non-common components are zero. To solve the explosion problem, we added the momenta vectors (product of mass and velocity) of two pieces to find the third one's momentum, keeping in mind the total resultant momentum should be zero to satisfy conservation laws.